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|1. How to calculate Cv homogeneity|
|2. Confidence intervals and how to assess a mix|
After sampling, the samples taken are analysed regarding the presence of the tracer. Then results are recorded and statistics can be peformed.
The mixture, regarding the tracer, will be defined according to the mean concentration of the tracer and to the standard deviation of the tracer concentration. Mean and standard deviation will be used to calculate the relative standard deviation of the mix, that will be the value assimilated to "homogeneity".
Equation 1 : Relative Standard Deviation of the mixture
S is the samples standard deviation, it is not the actual standard deviation since only an estimation can be given from the samples. μ is the arithemical average of the samples concentration, thus as well calculated from the samples.
This value is often expressed in % and called also the coefficient of variation
Equation 2 : Coefficient of variation of the mixture
The value obtained is then compared to the specification.
WARNING : the CV that will be obtained has actually several components, and some of these components need to be calculated in order to estimate the actual homogeneity variance.
The sample variance is calculated thanks to :
The variability due to sampling is very difficult to determine, thus, in practice, it will be neglected. However, for this assumption to be true it is critical to sample the mix following the methods explained above, and preferably on the free flowing powder.
The variability due to analysis can either be known, if experiments have been done before, or can be determined for the particular homogeneity validation by doubling the measurement on the same sample.
Smix can then be calculated. Then CVmix(%)
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WARNING : Once CVmix(%) calculated, it is not sufficient to compare it to the specification. Indeed, the variance calculated is NOT a true variance but is estimated on the sampling. This estimation must be taken into account by calculating a confidence interval, generally at 95%, which corresponds to 2 sigma on each side of the mean.
The specification can then be compared to the confidence interval
- If the CVspec > upper border of the confidence interval : the mixing is successful, the mix is significantly better than the specification
- If CVspec is in the confidence interval : the mixing may be successful, but it is also possible that the actual CVmix is > CVspec
- If CVspec is < lower border of the confidence interval : the mixing homogeneity achieved is not good enough for the application.